Is this a good enough reason to ask for an extension at university?

It'll have to be whether they like it or not. I'm sure any place of business or education has to make allowances for exceptional cirstances and this situation more than qualifies. They are people too and they should hopefully have the maturity to handle this difficult time for you with a professional conduct. Anyways I hope you are ok? I wish you all the best for you and your family.